In modern digital communication, a channel input signal is generally synthesized as a linear combination of certain bases functions whose coefficients bear information to be transmitted. For example, in an asymmetric digital subscriber line (ADSL) system using N sub-carriers, the basis function is represented by:
      x    ⁡          (      n      )        =            1      N        ⁢                  ∑                  k          =          0                          N          -          1                    ⁢                          ⁢                        X          ⁡                      (            k            )                          ⁢                  exp          ⁡                      (                                          j2                ⁢                                                                  ⁢                π                ⁢                                                                  ⁢                nk                            N                        )                              where X(k) is the coefficient bearing the information symbols to be transmitted. Since the inverse Fourier transform (IFT) is equivalent to a summation of several different frequencies, the resulting signal x(n) becomes approximately Gaussian for large N, according to the central-limit theorem. This means that there may be some very high peaks present in the signal x(n).
One of the problems encountered in the presence of high peaks within x(n) is a large peak-to-average ratio (PAR) value, or, equivalently, a large crest factor, which reduces the reliability of transmission systems. This is due to the fact that power amplifiers are typically only capable of modulating signals that are bounded by a fixed constant. Thus, any input signal exceeding this value is “clipped” at this level. In other words, if high peaks are present within x(n), then there is a possibility that these high peaks will be “clipped” because the high peaks exceed the bounds of the power amplifier. This introduces noise to the system, reduces the signal-to-noise ratio (SNR), and, thus, has strong impact on the reliability of the system. In order to reduce this effect, one can attenuate the amplitude of the entire signal. However, this worsens the SNR directly, and, further, reduces the system SNR due to increased quantization noise.
Given the problems associated with potentially large PAR values in digital communication systems, a need exists in the industry to reduce PAR values, thereby improving reliability of digital communication systems.